Polar Functions: Cheat Sheet

Essential Concepts

Polar Coordinates

Unlike the rectangular (Cartesian) coordinate system that uses perpendicular axes and coordinates [latex](x, y)[/latex], the polar coordinate system locates points using a distance from the origin and an angle from a fixed direction.

  • Components of the Polar System:
    • The pole: the origin point, corresponding to [latex](0, 0)[/latex] in rectangular coordinates
    • The polar axis: the reference direction (like the positive x-axis)
    • Distance [latex]r[/latex]: the radial distance from the pole
    • Angle [latex]\theta[/latex]: the direction measured counterclockwise from the polar axis
  • Plotting Polar Points [latex](r, \theta)[/latex]: Start at the pole, rotate counterclockwise by angle [latex]\theta[/latex] from the polar axis, then move distance [latex]r[/latex] in that direction. When [latex]r < 0[/latex], move in the opposite direction (rotate by [latex]\pi[/latex] and use [latex]|r|[/latex]).
  • Multiple Representations: A single point has infinitely many polar representations because angles can be coterminal ([latex](r, \theta) = (r, \theta + 2\pi k)[/latex]) and negative radius reverses direction ([latex](r, \theta) = (-r, \theta + \pi)[/latex]).
  • Converting Polar to Rectangular – Given [latex](r, \theta)[/latex], find [latex](x, y)[/latex]:
    • [latex]x = r\cos\theta[/latex]
    • [latex]y = r\sin\theta[/latex]
  • Converting Rectangular to Polar – Given [latex](x, y)[/latex], find [latex](r, \theta)[/latex]:
    • [latex]r = \sqrt{x^2 + y^2}[/latex] (always take positive [latex]r[/latex] when converting)
    • [latex]\tan\theta = \frac{y}{x}[/latex], then use [latex]\theta = \tan^{-1}(\frac{y}{x})[/latex]

Be careful with the angle [latex]\theta[/latex]. Check which quadrant the point [latex](x, y)[/latex] is in, as inverse tangent only gives values in quadrants I and IV. Adjust [latex]\theta[/latex] accordingly.

Transforming Equations Between Forms

  • [latex]x = r\cos\theta[/latex] and [latex]y = r\sin\theta[/latex]
  • [latex]r^2 = x^2 + y^2[/latex]
  • [latex]\tan\theta = \frac{y}{x}[/latex]
  • For rectangular to polar: substitute [latex]x = r\cos\theta[/latex] and [latex]y = r\sin\theta[/latex], simplify, and solve for [latex]r[/latex] in terms of [latex]\theta[/latex].
  • For polar to rectangular: replace [latex]r\cos\theta[/latex] with [latex]x[/latex] and [latex]r\sin\theta[/latex] with [latex]y[/latex], use [latex]r^2 = x^2 + y^2[/latex] when needed, then simplify.

Graphing in Polar Coordinates

  • Symmetry about [latex]\theta = \frac{\pi}{2}[/latex] (y-axis): Replace [latex](r, \theta)[/latex] with [latex](-r, -\theta)[/latex]. If equivalent, the graph has this symmetry.
  • Symmetry about polar axis (x-axis): Replace [latex](r, \theta)[/latex] with [latex](r, -\theta)[/latex] or [latex](-r, \pi - \theta)[/latex]. If equivalent, the graph has this symmetry.
  • Symmetry about the pole (origin): Replace [latex](r, \theta)[/latex] with [latex](-r, \theta)[/latex]. If equivalent, the graph has this symmetry.

Finding Key Features:

  • Zeros: Set [latex]r = 0[/latex] and solve for [latex]\theta[/latex]
  • Maximum [latex]|r|[/latex]: Find when the trig function reaches its maximum value (1 for sine/cosine)
  • Intercepts: Evaluate [latex]r[/latex] at [latex]\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}[/latex]

Classic Polar Curves:

Circles:

  • [latex]r = a\cos\theta[/latex]: circle with diameter [latex]|a|[/latex], centered at [latex](\frac{a}{2}, 0)[/latex]
  • [latex]r = a\sin\theta[/latex]: circle with diameter [latex]|a|[/latex], centered at [latex](0, \frac{a}{2})[/latex]

Cardioids (heart-shaped):

  • Forms: [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex] where [latex]\frac{a}{b} = 1[/latex] ([latex]a > 0, b > 0[/latex])
  • Passes through the pole

Limaçons (snail-shaped):

  • Forms: [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex]
  • Dimpled (one loop with dimple): [latex]1 < \frac{a}{b} < 2[/latex]
  • Convex (no dimple): [latex]\frac{a}{b} \geq 2[/latex]
  • Inner-loop: [latex]a < b[/latex] (creates a loop inside)

Lemniscates (figure-eight or infinity symbol):

  • Forms: [latex]r^2 = a^2\cos(2\theta)[/latex] or [latex]r^2 = a^2\sin(2\theta)[/latex]
  • Always symmetric with respect to the pole
  • Cosine version is also symmetric about polar axis and [latex]\theta = \frac{\pi}{2}[/latex]

Rose Curves (flower petals):

  • Forms: [latex]r = a\cos(n\theta)[/latex] or [latex]r = a\sin(n\theta)[/latex]
  • If [latex]n[/latex] is even: curve has [latex]2n[/latex] petals
  • If [latex]n[/latex] is odd: curve has [latex]n[/latex] petals
  • Each petal has length [latex]|a|[/latex]

Archimedes’ Spiral:

  • Form: [latex]r = \theta[/latex] (for [latex]\theta \geq 0[/latex])
  • Distance from pole increases at constant rate as the curve spirals outward

Polar Form of Complex Numbers

  • The complex plane plots complex number [latex]z = a + bi[/latex] with real part [latex]a[/latex] on the horizontal axis and imaginary part [latex]b[/latex] on the vertical axis.
  • Absolute Value (Modulus): [latex]|z| = \sqrt{a^2 + b^2}[/latex]
  • Polar Form: [latex]z = r(\cos\theta + i\sin\theta) = r\text{ cis }\theta[/latex]. Where [latex]r = |z| = \sqrt{a^2 + b^2}[/latex] (the modulus) and [latex]\theta[/latex] is the argument (angle from positive real axis). We have [latex]\cos\theta = \frac{a}{r}[/latex] and [latex]\sin\theta = \frac{b}{r}[/latex].
    • Converting Rectangular to Polar: Find [latex]r = \sqrt{a^2 + b^2}[/latex], find [latex]\theta[/latex] using [latex]\cos\theta = \frac{a}{r}[/latex] or [latex]\tan\theta = \frac{b}{a}[/latex] (watch the quadrant), then write as [latex]r(\cos\theta + i\sin\theta)[/latex].
    • Converting Polar to Rectangular: Evaluate [latex]\cos\theta[/latex] and [latex]\sin\theta[/latex], calculate [latex]a = r\cos\theta[/latex] and [latex]b = r\sin\theta[/latex], then write as [latex]a + bi[/latex].
  • Operations in Polar Form:
    • Product – Multiply moduli, add angles: [latex]z_1 z_2 = r_1 r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)][/latex]
    • Quotient – Divide moduli, subtract angles: [latex]\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)][/latex]
    • De Moivre’s Theorem (for powers): [latex]z^n = r^n[\cos(n\theta) + i\sin(n\theta)][/latex]. Raise the modulus to the [latex]n[/latex]th power and multiply the angle by [latex]n[/latex].
    • nth Root Theorem (for roots): [latex]z^{\frac{1}{n}} = r^{\frac{1}{n}}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right][/latex]. Where [latex]k = 0, 1, 2, ..., n-1[/latex] gives all [latex]n[/latex] distinct roots. Add [latex]\frac{2k\pi}{n}[/latex] to [latex]\frac{\theta}{n}[/latex] to obtain each root.

Key Equations

Polar to rectangular [latex]x = r\cos\theta[/latex]

[latex]y = r\sin\theta[/latex]
Rectangular to polar  [latex]r = \sqrt{x^2 + y^2}[/latex]

[latex]\tan\theta = \frac{y}{x}[/latex]
Circle (polar form) [latex]r = a\cos\theta[/latex]

[latex]r = a\sin\theta[/latex]
Cardioid [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex]
where [latex]\frac{a}{b} = 1[/latex]
Limaçon [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex]
where [latex]\frac{a}{b} \neq 1[/latex]
Lemniscate [latex]r^2 = a^2\cos(2\theta)[/latex] or [latex]r^2 = a^2\sin(2\theta)[/latex]
Rose curve [latex]r = a\cos(n\theta)[/latex] or [latex]r = a\sin(n\theta)[/latex]
Archimedes’ spiral [latex]r = \theta[/latex]
Complex number absolute value [latex]|z| = \sqrt{a^2 + b^2}[/latex]
Complex number polar form [latex]z = r(\cos\theta + i\sin\theta) = r\text{ cis }\theta[/latex]
Product of complex numbers [latex]z_1 z_2 = r_1 r_2\text{ cis}(\theta_1 + \theta_2)[/latex]
Quotient of complex numbers [latex]\frac{z_1}{z_2} = \frac{r_1}{r_2}\text{ cis}(\theta_1 - \theta_2)[/latex]
De Moivre’s Theorem [latex]z^n = r^n[\cos(n\theta) + i\sin(n\theta)][/latex]
nth Root Theorem [latex]z^{\frac{1}{n}} = r^{\frac{1}{n}}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right][/latex]
where [latex]k = 0, 1, 2, ..., n-1[/latex]

Glossary

absolute value of a complex number

The distance from the origin to the point representing the complex number in the complex plane; for [latex]z = a + bi[/latex], it equals [latex]\sqrt{a^2 + b^2}[/latex].

Archimedes’ spiral

A polar curve with equation [latex]r = \theta[/latex] that spirals outward from the pole with distance increasing at a constant rate.

argument

The angle [latex]\theta[/latex] in the polar form of a complex number, measured from the positive real axis.

cardioid

A heart-shaped polar curve with equation [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex] where [latex]\frac{a}{b} = 1[/latex].

complex plane

A coordinate plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers; used to plot complex numbers.

De Moivre’s Theorem

A formula for raising a complex number in polar form to a power: [latex]z^n = r^n[\cos(n\theta) + i\sin(n\theta)][/latex].

lemniscate

A figure-eight shaped polar curve with equation [latex]r^2 = a^2\cos(2\theta)[/latex] or [latex]r^2 = a^2\sin(2\theta)[/latex].

limaçon

A snail-shaped family of polar curves with equation [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex]; includes dimpled, convex, and inner-loop varieties depending on the ratio [latex]\frac{a}{b}[/latex].

modulus

The absolute value or magnitude of a complex number; the distance [latex]r[/latex] from the origin in polar form.

polar axis

The reference direction in the polar coordinate system, corresponding to the positive x-axis; the starting line from which angles are measured.

polar coordinates

An ordered pair [latex](r, \theta)[/latex] where [latex]r[/latex] is the distance from the pole and [latex]\theta[/latex] is the angle from the polar axis.

polar form of a complex number

The representation [latex]z = r(\cos\theta + i\sin\theta)[/latex] or [latex]r\text{ cis }\theta[/latex], where [latex]r[/latex] is the modulus and [latex]\theta[/latex] is the argument.

pole

The origin of the polar coordinate system, corresponding to the point [latex](0, 0)[/latex] in rectangular coordinates.

rose curve

A petal-shaped polar curve with equation [latex]r = a\cos(n\theta)[/latex] or [latex]r = a\sin(n\theta)[/latex]; has [latex]n[/latex] petals if [latex]n[/latex] is odd, [latex]2n[/latex] petals if [latex]n[/latex] is even.

symmetry with respect to the polar axis

A graph is symmetric about the x-axis if replacing [latex](r, \theta)[/latex] with [latex](r, -\theta)[/latex] yields an equivalent equation.

symmetry with respect to the pole

A graph is symmetric about the origin if replacing [latex](r, \theta)[/latex] with [latex](-r, \theta)[/latex] yields an equivalent equation.

symmetry with respect to [latex]\theta = \frac{\pi}{2}[/latex]

A graph is symmetric about the y-axis if replacing [latex](r, \theta)[/latex] with [latex](-r, -\theta)[/latex] yields an equivalent equation.